d
(d)

Work done by the gravitational force acting on the particle \(D\) during its movement

\(=-\Delta U\)

\(=-\left(U_{\text {final }}-U_{\text {initial }}\right)\)

\(=U_{\text {initial }}-U_{\text {final }}\)

Now, when the particle is at infinity, \(U=0\)

\(\Rightarrow U_{\text {final }}=0\)

\(\Rightarrow \text { Work done }=U_{\text {initial }}\)

\(U_{\text {initial }}=-\frac{G m^2}{L}-\frac{G m^2}{L}-\frac{G m^2}{\sqrt{2} L}\)

\(=-\frac{G m^2}{L}\left(2+\frac{1}{\sqrt{2}}\right)\)

\(=-\frac{G m^2}{L}\left(\frac{2 \sqrt{2}+1}{\sqrt{2}}\right)\)